Integrator



July 16, 1946. w. B. JORDAN INTEGRATOR Filed Sept. 18, 1944 Hf A *Y /l- Inventor* William B. by yv/Mwq JoT' dan His Attorneg.

Patented July 16, 1946 IN TEGRATOR William B. Jordan, Scotia, N. Y., assigner to General Electric Company, a corporation of New York Application September 18, 1944, Serial No. 554,661

(Cl. 23S-61) 4 Claims. 1

This invention relates to integrators, and it has for its object the provision of an improved integrator which is positive and accurate in its operation.

More specifically, this invention contemplates the provision of a mechanical integrator which computes its integral by the trapezodal rule; that is, it mechanically computes the desired area under a curve by sub-dividing the area into a series of trapezoids; computing the area of each trapezoid by multiplying one-half the sum of the lengths of two sides thereof by the base length; and finally by adding the calculated areas of the trapezoids together.

In accordance with this invention, there is provided an integrand input shaft and a variable of integration input shaft. Means are provided for sub-dividing the total variable of integration input to determine the interval of integration. And also means are provided for determining the product of this interval and the values of the integrand introduced by the integrand input shaft so as to determine the integral of the integrand with reference to the variable of integration,

More specifically, a first reverser is provided which is operated by the variable of integration input shaft; it periodically reverses the direction of rotation of the input thereto, and thereby subdivides the total of the variable of integration input in order to determine the interval of integration. This interval is multiplied by the integrand values fed in by the integrand input shaft by a suitable multiplying mechanism in order to determine the areas of the various subdivisions. The output of the multiplier is fed to an output shaft which, because of the reversals of the reversing mechanism ,is driven first in one direction and then in the other, and the total motion of the shaft is imparted to a final and integral output shaft by means of a second reverser. This second reverser always operates the final shaft in one direction, until the integrand changes sign, so that the total movement of the shaft is the combined forward and backward movements of the multiplier output shaft.

For a more complete understanding of this invention, reference should be had to the accompanying drawing in which Fig. 1 is a perspective view illustrating an integrator arranged in accordance with this invention; and Fig. 2 is a diagrammatic view illustrating the principle of operation of the mechanism.

Referring to the drawing, this invention has been shown in one form as applied to an integrator comprising a variable o1' integration input shaft I0, an integrand input shaft II, and a final integral output shaft I2. The specific einbodiment of the invention illustrated integrates with respect to time (t) the magnitude of a variable rate of change which may be the rate of change of range to a target. Thus, the variable of integration (t) is introduced by the shaft I0, the integrand is introduced by the shaft Il, while the final output of the integral shaft I2 is (y), (y) being related to a: and t-by the equation: y=lfxdt.

The shaft I0 operates a first reversing mechanism I3; as shown, the shaft I0 is geared to drive an input gear I4 of the reverser I3 by means of a shaft I5 to which the shaft I0 is geared by bevel gears I6, and by a shaft I'I to which the gear I4 is secured and which shaft is geared to shaft I5 by bevel gears I8. The gear I4 is in mesh with gears I9 and 2U which it rotates in opposite directions and which are mounted on a shaft 2I to rotate freely with reference to it. A shuttle 22, which is slidably mounted on the shaft 2I but which is arranged to rotate it, as by means of a splined connection (not shown), is arranged to be operated selectively by the two reversely rotating gears I9 and 20; for this purpose, the shuttle is provided on its opposite ends with clutch teeth 23 and 24 which are adapted to mesh with sets of teeth 23a and 24a. `formed on the gears I9 and 20, respectively. The shuttle is shown in its neutral position in Fig. 1, and when moved upwardly to cause its tooth 23 to engage gear tooth 23a it causes the shaft 2| to rotate in one direction, and when moved downwardly to cause its tooth 24 to mesh with the gear teeth 24a it causes the shaft 2| to rotate in the opposite direction.

The shuttle is moved to its upper and lower driving means periodically by means of a cam 25 which operates between a pair of spaced collars 25a and 25D on the shuttle. This cam 25 is periodically operated by the input shaft I0. This is accomplished through a Geneva gear train 26 having three stages 26a, 2Gb, and 26e. This gear train is, in effect, a revolution counter, and every time the shaft l0 turns over a predetermined number of revolutions the cam 25 is operated to move the shuttle to reverse the output of the reverser mechanism; as shown, the output of the gear 21 of the Geneva train meshes with a gear 28 which is fixed to and drives a gear 29. The gear unit 28 and 29 rotates freely on the shaft I0. The gear 29 drives a gear 30 which in turn operates the shuttle cam 25 through a shaft 3l,

a mi

bevel gears 32, shaft 3f, bevel gears 34 and cam shaft In the specific embodiment of the invention illustrated where the input of the shaft is is time (t), the mechanism operates the shuttle everyT (it) seconds, and therefore, in effect, mechanically sub-divides the total variable of integration (t) in order to determine the interval of integration (t).

The output of the reverser i3 is added to the input (r) of the integrand input shaft l l in a differential 36. This differential, as shown, has an input gear 3l, which is driven by gear 38 attached to the shaft 2l; an input gear 39 driven by a gear iii attached to the shaft il and a set of planetary gears lli which drive the output shaft 2 of the differential, which output is the sum` mation:

t of shaft +r of shaft 11) Another similar differential 43 subtracts the input (x) from input (t). This differential has an input gear filldriven by the output gear-3B of reverser i3 through the gear 3l; an input gear l5 driven by the gear il@ of input shaft Il; and a planetary gear set e6 which drives the differential output shaft il to measure the difference t-.

The output shafts @2 and 4l of the two differentials 36 and i3 are fed into two cam mechanisms fill and i9 respectively, which mechanisms generate quadratic functions of the inputs. The two mechanisms i8 and i9 comprise cams 5!! and El respectively having formed therein a series of gear slots 52 and 53 arranged in spiral paths, and with which toothed gear wheels 54 and 55 mesh respectively. The cam plate 50 has a peripheral gear 5t which meshes with gear El driven by shaft i2 while cam 5l has a peripheral gear driven by gear 59 of shaft lll.

Therefore, `the summation of t|-5c is fed into gear cam 5&3, while the difference t-:r is fed into gear cam 5l.

In order to always keep positive the radii of the two spiral paths of gear slots 52 and 53, a constant (K) is added to each input so that the input to the cam 50 becomes t-i-K-l-, while that to cam 5l becomes t-i-K-au It will be understood that this constant (K) is generated by offsetting the Zero position between the outputs of the differentials 36 and i3 and two cams 50 and fil respectively.

As pointed out above, the two mechanisms 43 and it generate quadratic functions of their inputs; in the specic embodiment of the invention illustrated, they generate the squares of their inputs; thus, the mechanism i8 generates and its output shaft dfi measures (t-l-K-i-)2, and the mechanism 9 generates and its output shaft 5l measures (t+K-;c)2. These two outputs are subtracted in a differential Gla.

This differential lila, as shown, has an input gear 62 driven by gear h3 fixed to shaft 60; an input gear Se driven by gear 65 fixed to shaft el, and a planetary gear system 66 having a ring gear Gl, which measures the aforementioned difference (t-i-K-Hw 2-(t-{Kx)2; this difference equals: ext-PMK or 4MM-K) The constant K is deleted in a differential 68.

by subtracting 1Km from HUH-K) so that the output of the differential 68 is the product ext. As shown, the differential 63 has a planetary gear system e9 which includes a ring gear 70 which meshes with output ring gear 6l of differential te; in otherwords, 4r(t-|-.K) is introduced into i differential 63 by ring gear 6l. Differential 69 also has an input gear 'il which is driven by shaft El 'to introduce elim; as shown, it is driven from shaft li by bevel gears l2, shaft "i3, spur gear 'M fixed to shaft f3 and an idler gear l meshing with the gears 'El and lli. The gear ratio between shaft l i and gear H is such that the rotation of the gear li measures the product li-Kaz. This product is subtracted from the product 4MM-K) in the differential 6% so that the differential output gear l measures 43st.

It will be observed, in View of the foregoing, that the differentials if and (ai, the two cam mechanisms and ff, and the two differentials and it constitute a multiplier wherein the inputs thereto from the reverser i3 and the input shaft il are multiplied so that the output becomes :the product 40st.

The output fait of the multiplier is fed to a shaft 'Il from gear it, which gear meshes with gear lla, fixed to the shaft 'V so that the motion of the shaft il measures the product 496i.

The shaft l? is utilized to drive the nal integral output shaft l2. Because of the action of the reverser E3 in periodically reversing, the motion of the shaft ll is rst positive and then negative. It is desirable that the final output shaft if always be driven in the same direction, and that it add up the total motions of the shaft 'il in its two directions; therefore, the shaft 'Il drives `the final output shaft l2 through a second reverser TES. This reverser comprises input gears le and 88 which are driven in opposite directions by the shaft 'if-the gear l@ by means of gear 8l xed to the shaft il, and the gear 86 from the gear 8l through gear 32 fixed to a shaft 83, and a gear Sit also fixed to the shaft 83 and meshing with the gear 8f?, This reverser further comprises a shuttle 85 with two sets of teeth 8l' and e3 which are arranged to mesh with sets of teeth 3S and 89 formed on the two input gears F9 and e@ respectively- The shuttle is splined to the shaft i2 and, therefore, when it rotates it rotates this shaftl The position of the shuttle 85 is controlled by means of a cam 9E! that operates between the shuttle collars l and 92. The cam is driven by the shaft 3i so that the reverser 'I8 operates to reverse periodically and simultaneously with the reversals of the reverser I3, whereby even though the output shaft l? of the multiplier moves first in one direction and then in the other in accordance with the product 43st, the nal output shaft i2 always moves in the same direction and measures the total motions of the shaft l1.

And this nal output of g/:fdt

The theory of operation of this integrator may be better understood by reference to Fig. 2. The total area under the curve, which represents the equation x= t), is equal to the sum of 'the subdivided areas of the trapezoids S1, S2, etc. The area of the shaft l2 that of S2=1/2(x1}$2)h and so on.

Now at the instant of time tu the instantaneous position of the output shaft 2l of the reverser i3 is -l/gh and the reverser has just thrown over to its forward position; and at the instant of time t1, the instantaneous position the output shaft 2l is -l-l/ah and the reverser throws to its reverse position and reverses its output; the

zero position of the shaft 2| corresponds to the time point exactly mid-way between to and t1 i11- dicated by the vertical dotted line in Fig. 2.

When the shaft 2| is at position -1/2h, the :n input shaft I I is at its position o, and the output shaft 'I'I of the multiplying mechanism is at its position l/zhxo.

Then at the end of one-half of an operating cycle of the mechanism, that is, at the end of the time interval h=t1-o, the position of shaft 2i is +1/ h, that of shaft II is an, and that of shaft 'I1 is -i-l/ghsci.

The net rotation of the shaft I7 at the end of the first half cycle, that is, at the end of time t1-to, is the difference betweenits position at the end of the first half cycle and its position at the beginning of the rst half cycle, or

Thus, during the first half cycle, the shaft 2I moves from its initial forward position corresponding to Mah to its extremefposition in its forward motion corresponding to -i-/gh, and as a result the movement of shaft 'I'I is 1/2(o-|sci)h.

During the next half cycle the shaft 2I moves in the reverse direction, and it moves from its position +1/2h, back to its position -1/271, and the position of input shaft a: changes from :ci to x2. The position of output shaft 'I'I at the start cf the second half cycle is -I-/ghi and at the end thereof is -l/ghz, and the net change in the position of the shaft 11 during the second half cycle is:

During all of this time, the final output shaft I2 is operated by the shaft 'I'I through the reverser 18, this reverser always functioning to add the total forward and backward movements of the shaft 'I'I and imparting the sum te' the shaft I2. Therefore, during the first half cycle when the reverser I3 is in its forward motion, the change in position of shaft I2 is the same as that of shaft 11, that is, it is equal to 1/2(:vo+i)h; however, the change in position of the final shaft I2 during the second half cyclewhen the reverser 'I8 has reversed its operationis the negative of the change in position of the shaft 'I'I during the second half cycle, that is, it is the negative of -1/2h(:r1-I-:rz) which, of course, is +1/2h(:r1-I2).

Therefore, the total change in the position of the final output shaft I2 from time to to time t2 is Immediately upon the completion of the first cycle, both reversers I3 and 18 operate to reverse and the complete cycle is repeated. And in this way the mechanism adds up the areas of all of the trapezoids under the curve. In other Words, the total motion of the final output shaft I2 measures the areas of all the trapezoids under the curve, and consequently measures the desired integral.

It will be understood that this integrator will operate to calculate the integral of x regardless of whether :c is positive or negative.

The description of this integrator and of its operation given above show how integrals may be calculated by this integrator using the trapezoidal rule. It is also possible, however, to use it for computing integrals by Simpsons onethird rule or by other well-known mechanical quadrature formulas by an adjustment of the Zero position of the multiplier. For example, if the shaft 2I is at position -lgh when the .7: input shaft II is at its position ro, and the reversers I3 and 'I8 have just thrown to their forward driving positions at this instant, then the integrator will calculate the integral fdt based on Simpsons one-third rule.

While I have shown a particular embodiment of my invention, it will be understood, of course, that I do not wish to be limited thereto since many modifications may be made, and I, therefore, contemplate by the appended claims to cover any such modifications as fall within the true spirit and scope of my invention.

What I claim as new and desire to secure by Letters Patent of the United States, is:

1. An integrator comprising variable of integration input means, integrand input means, reversing mechanism operable in accordance with the operation of said variable of integration input means for periodically reversing the direction of the input, means for multiplying the output of said reversing mechanism by the input of said integrand input means, and means for periodically reversing the direction of the output of said multiplying means simultaneously with the reversals of said rst reversing mechanism.

2. An integrator comprising a variable of integration input shaft, a reversing mechanism operated by said input shaft constructed and arranged to reverse periodically the direction of rotation of its output with reference to the input thereto, an integrand input shaft, a multiplier operated by the output of said reversing mechanism and the input of said integrand input shaft operating to multiply said output and input together, a final integral output shaft, and a second reversing mechanism operating said final output shaft and controlled to reverse periodically and simultaneously with said first reverser so as to periodically reverse the output of said multiplier to impart its motions to said nal output shaft.

3. An integrator comprising a variable of integration input shaft, an integrand input shaft, an integral output shaft, a first reverser having an output shaft, said reverser being operated by said variable of integration input shaft so that said output shaft is periodically reversed with reference to said variable of integration input shaft, the output shaft thereby measuring the interval of integration, first mechanism for adding the output of said output shaft with the input of said integrand input shaft and for squaring said summation, second mechanism for subtracting from the output of said output shaft from the input of said integrand input shaft and for squaring the difference, means for subtracting the output of said second mechanism from that of said first mechanism so as to obtain the product of the variable of integration input and the integrand input, an output shaft driven first in one direction and then in the other by the output of said last-named means, and a second reverser for operating said integral output shaft operated by said variable of integration input shaft to reverse periodically and simultaneously with said first reverser so that the total motion of said last-named output shaft in both directions is combined as a motion in said final integral output shaft.

4. An integrator comprising a variable of `integration rinput shaft, an integrand input shaft, an integral output shaft, a first reverser having an output shaft, said reverser being operated by said variable of integration input shaft so that said output shaf t is periodically reversed with reference to said variable of integration input shaft, the output shaft thereby measuring the interval of integration, a differential having an input driven by said output shaft of said reverser, a second differential having an input driven by said output shaft of said reverser, said two differentials also having inputs operated by said integrand input shaft, and the two differentials also having output shafts, the first dierential however generating and delivering to its output shaft the summation of the reverser output and the input of said integrand shaft, while the second generates and delivers to its output shaft the difference between the output of said reverser and the input of said integrand shaft, a first cam mechanism operated by the output of said first differential for generating the square of said summation, a second cam mechanism operated by the output of said second differential for generating the square of said difference, a third differential operated by said two cam mechanisms having' an output which generates the difference between the outputs of said first and second cam mechanisrns to thereby generate the product of the integrand input and the variable of integration input, a'shaft driven by the output of said third differential, a second reverser controlled by the operation of said variable of integration input shaft so as to reverse periodically and simultaneously with the reversals of said first reverser, and a driving connection between said second reverser and said integral output shaft so that said shaft moves through the total distance that said last-named shaft is driven by said third differential.

WILLIAM B, JORDAN. 

